A new family of second-order absorbing boundary conditions for the acoustic wave equation - Part II : Mathematical and numerical studies of a simplified formulation

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A new family of second-order absorbing boundary conditions for the acoustic wave equation - Part II :

Mathematical and numerical studies of a simplified formulation

Hélène Barucq, Julien Diaz, Véronique Duprat

To cite this version:

Hélène Barucq, Julien Diaz, Véronique Duprat. A new family of second-order absorbing boundary

conditions for the acoustic wave equation - Part II : Mathematical and numerical studies of a simplified

formulation. [Research Report] RR-7575, INRIA. 2011. �inria-00578152�

a p p o r t

d e r e c h e r c h e

0249-6399ISRNINRIA/RR--7575--FR+ENG

Observation and Modeling for Environmental Sciences

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A new family of second-order absorbing boundary conditions for the acoustic wave equation - Part II : Mathematical and numerical studies of a simplified

formulation

Hélène Barucq — Julien Diaz — Véronique Duprat

N° 7575

March 2011

Centre de recherche INRIA Bordeaux – Sud Ouest

Domaine Universitaire - 351, cours de la Libération 33405 Talence Cedex

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❛❜s♦r❜✐♥❣

❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡

❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✿ ▼❛t❤❡♠❛t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧

st✉❞✐❡s ♦❢ ❛ s✐♠♣❧✐✜❡❞ ❢♦r♠✉❧❛t✐♦♥

❍é❧è♥❡ ❇❛r✉❝q

✱ ❏✉❧✐❡♥ ❉✐❛③

✱ ❱ér♦♥✐q✉❡ ❉✉♣r❛t

❚❤❡♠❡ ✿ ❖❜s❡r✈❛t✐♦♥ ❛♥❞ ▼♦❞❡❧✐♥❣ ❢♦r ❊♥✈✐r♦♥♠❡♥t❛❧ ❙❝✐❡♥❝❡s

➱q✉✐♣❡s✲Pr♦❥❡ts ▼❆●■◗❯❊✲✸❉

❘❛♣♣♦rt ❞❡ r❡❝❤❡r❝❤❡ ♥➦ ✼✺✼✺ ✖ ▼❛r❝❤ ✷✵✶✶ ✖ ✸✾ ♣❛❣❡s

❆❜str❛❝t✿ ❲❡ ✐♥t❡r❡st ♦✉rs❡❧✈❡s ♦♥ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❞ t❤❡ ♥✉♠❡r✐❝❛❧ ♣r♦♣✲

❡rt✐❡s ♦❢ ❛ s✐♠♣❧✐✜❡❞ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❛ ♥❡✇ ❢❛♠✐❧② ♦❢ ❛❜s♦r❜✐♥❣ ❜♦✉♥❞❛r② ❝♦♥✲

❞✐t✐♦♥s ✭❆❇❈s✮ ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ✜rst ♣❛rt ♦❢

t❤✐s ✇♦r❦✳ ❈♦♥s✐❞❡r✐♥❣ ❛ s♦✉♥❞✲s♦❢t s❝❛tt❡r❡r✱ ✇❡ ♣r♦✈❡ t❤❡ ✇❡❧❧✲♣♦s❡❞♥❡ss ♦❢

t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦❢ t❤❡

s♦❧✉t✐♦♥✳ ❲❡ ♣❡r❢♦r♠ ❛ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s✳ ❲❡ ♣r♦♣♦s❡ t♦ ✉s❡ ❛ ❉✐s❝♦♥t✐♥✉♦✉s

●❛❧❡r❦✐♥ ♠❡t❤♦❞ ❢♦r t❤❡ s♣❛❝❡ ❞✐s❝r❡t✐③❛t✐♦♥ ❛♥❞ ❜② ❞❡✜♥✐♥❣ ❛ ❞✐s❝r❡t❡ ❡♥❡r❣②

✇❡ ♣r♦✈❡ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡✳ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❝♦♥✜r♠ t❤❡

t❤❡♦r❡t✐❝❛❧ ♣r♦♣❡rt✐❡s ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ ❛♥❞ ✐❧❧✉str❛t❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡s ♦❢ t❤❡

♥❡✇ ❆❇❈s✳

❑❡②✲✇♦r❞s✿ ❆❜s♦r❜✐♥❣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥✱ ✇❡❧❧✲

♣♦s❡❞♥❡ss✱ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛②✱ ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ♠❡t❤♦❞

∗❚❡❛♠✲♣r♦❥❡❝t ■◆❘■❆ ▼❆●■◗❯❊✲✸❉

†▲▼❆P✱ ❯♥✐✈❡r❡s✐t② ♦❢ P❛✉

❯♥❡ ♥♦✉✈❡❧❧❡ ❢❛♠✐❧❧❡ ❞❡ ❝♦♥❞✐t✐♦♥s ❛✉① ❧✐♠✐t❡s

❛❜s♦r❜❛♥t❡s ♣♦✉r ❧✬éq✉❛t✐♦♥ ❞❡s ♦♥❞❡s

❛❝♦✉st✐q✉❡s ✲P❛rt✐❡ ■■ ✿ ❊t✉❞❡s ♠❛t❤é♠❛t✐q✉❡ ❡t

♥✉♠ér✐q✉❡ ❞✬✉♥❡ ❢♦r♠✉❧❛t✐♦♥ s✐♠♣❧✐✜é❡

❘és✉♠é ✿ ◆♦✉s ♥♦✉s ✐♥tér❡ss♦♥s ❛✉① ♣r♦♣r✐étés ♠❛t❤é♠❛t✐q✉❡s ❡t ♥✉♠ér✐q✉❡s

❞✬✉♥❡ ❢♦r♠✉❧❛t✐♦♥ s✐♠♣❧✐✜é❡ ❞✬✉♥❡ ♥♦✉✈❡❧❧❡ ❢❛♠✐❧❧❡ ❞❡ ❝♦♥❞✐t✐♦♥s ❛✉① ❧✐♠✐t❡s

❛❜s♦r❜❛♥t❡s ✭❈▲❆s✮ ♣♦✉r ❧✬éq✉❛t✐♦♥ ❞❡s ♦♥❞❡s ❛❝♦✉st✐q✉❡s ✐♥tr♦❞✉✐t❡ ❞❛♥s ❧❛

♣r❡♠✐èr❡ ♣❛rt✐❡ ❞❡ ❝❡ tr❛✈❛✐❧✳ ❊♥ ❝♦♥s✐❞ér❛♥t ✉♥❡ ❝♦♥❞✐t✐♦♥ ❞❡ s✉r❢❛❝❡ ❧✐❜r❡ s✉r

❧❡ ❜♦r❞ ❞❡ ❧✬♦❜st❛❝❧❡✱ ♦♥ ♠♦♥tr❡ q✉❡ ❧❡ ♣r♦❜❧è♠❡ ❛✉① ❧✐♠✐t❡s ❝♦rr❡s♣♦♥❞❛♥t ❡st

❜✐❡♥ ♣♦sé ❡t q✉❡ ❧❛ s♦❧✉t✐♦♥ ❞é❝r♦ît ❡①♣♦♥❡♥t✐❡❧❧❡♠❡♥t✳ ❖♥ ré❛❧✐s❡ ❛✉ss✐ ✉♥❡

❛♥❛❧②s❡ ♥✉♠ér✐q✉❡✳ ❖♥ ♣r♦♣♦s❡ ❞✬✉t✐❧✐s❡r ✉♥❡ ♠ét❤♦❞❡ ❞❡ ●❛❧❡r❦✐♥❡ ❞✐s❝♦♥✲

t✐♥✉❡ ♣♦✉r ❧❛ ❞✐s❝rét✐s❛t✐♦♥ ❡♥ ❡s♣❛❝❡ ❡t ❡♥ ❞é✜♥✐ss❛♥t ✉♥❡ é♥❡r❣✐❡ ❞✐s❝rèt❡✱ ♦♥

♠♦♥tr❡ ❧❛ st❛❜✐❧✐té ❞✉ s❝❤é♠❛ ♥✉♠ér✐q✉❡✳ ❉❡s rés✉❧t❛ts ♥✉♠ér✐q✉❡s ❝♦♥✜r♠❡♥t

❧❡s ♣r♦♣r✐étés t❤é♦r✐q✉❡s ♦❜t❡♥✉❡s ❡t ✐❧❧✉str❡♥t ❧❡s ♣❡r❢♦r♠❛♥❝❡s ❞❡s ♥♦✉✈❡❧❧❡s

❈▲❆s✳

▼♦ts✲❝❧és ✿ ❈♦♥❞✐t✐♦♥s ❛✉① ❧✐♠✐t❡s ❛❜s♦r❜❛♥t❡s✱ éq✉❛t✐♦♥ ❞❡s ♦♥❞❡s ❛❝♦✉s✲

t✐q✉❡s✱ ♣r♦❜❧è♠❡ ❜✐❡♥ ♣♦sé✱ ❞é❝r♦✐ss❛♥❝❡ ❡①♣♦♥❡♥t✐❡❧❧❡✱ ♠ét❤♦❞❡ ❞❡ ●❛❧❡r❦✐♥❡

❞✐s❝♦♥t✐♥✉❡

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✸

✶ ■♥tr♦❞✉❝t✐♦♥

❚❤✐s ✇♦r❦ ❢♦❧❧♦✇s ❛ ♣r❡✈✐♦✉s ♦♥❡ ✐♥ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ❛ ♥❡✇ ❢❛♠✐❧②

♦❢ ❆❇❈s ❞❡♣❡♥❞✐♥❣ ♦♥ ❛ ♣❛r❛♠❡t❡r✳ ❚❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡

♠✐❝r♦✲❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥✱ ❢♦❧❧♦✇✐♥❣ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢

❢❛❝t♦r✐③❛t✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ ❬✶✺❪ ❛♥❞ ❢♦r♠❡r❧② ✉s❡❞ ❜② ❊♥❣q✉✐st ❛♥❞ ▼❛❥❞❛ ❬✾❪

❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❆❇❈s ❢♦r ✢❛t s✉r❢❛❝❡s✳ ❲❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ s❡❝♦♥❞✲♦r❞❡r

❝♦♥❞✐t✐♦♥s ✇❤✐❝❤ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ♦♥ ❛r❜✐tr❛r✐❧② s❤❛♣❡❞ s✉r❢❛❝❡s ❛♥❞ ✐♥ ❬✻❪✱ ✇❡

❤❛✈❡ s❤♦✇♥ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞✳ ◆♦✇✱

r❡❣❛r❞✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥s✱ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤❡s❡

❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ ✐♥❝❧✉❞❡❞ ✈❛r✐❛t✐♦♥❛❧② ♦♥❧② ✐❢ ✇❡ ❝♦♥s✐❞❡r ❛ ❤✐❣❤✲♦r❞❡r ❢✉♥❝✲

t✐♦♥❛❧ t❤❛t ✐s ♦❜t❛✐♥❡❞ ❛❢t❡r ❞❡r✐✈✐♥❣ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤✐s ♠✐❣❤t

❧❡❛❞ t♦ ❝r❡❛t❡ s♣✉r✐♦✉s s♦❧✉t✐♦♥s ❜❡❝❛✉s❡ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✲

✐♥❣ s②st❡♠ ❛❞♠✐ts st❛t✐♦♥♥❛r② s♦❧✉t✐♦♥s t❤❛t ❛r❡ ♥♦♥ ♣❤②s✐❝❛❧✳ ❆♥ ❛❧t❡r♥❛t✐✈❡

❝♦♥s✐sts ✐♥ ✐♥tr♦❞✉❝✐♥❣ ❛♥ ❛✉①✐❧✐❛r② ✉♥❦♥♦✇♥✳ ❇② t❤✐s ✇❛②✱ t❤❡ ♥❡✇ ❝♦♥❞✐t✐♦♥s

❝❛♥ ❜❡ ✐♥❝❧✉❞❡❞ ✈❛r✐❛t✐♦♥❛❧②✳ ❚❤❡ ❛✐♠ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ st✉❞② t❤❡ ❜♦✉♥❞❛r②

✈❛❧✉❡ ♣r♦❜❧❡♠ t❤❛t ✐s ♦❜t❛✐♥❡❞ ❜② ✐♥tr♦❞✉❝✐♥❣ ❛♥ ❛✉①✐❧✐❛r② ✉♥❦♥♦✇♥✳ ❲❡ ♣r♦✈❡

t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ❛♥❞ ✇❡ s❤♦✇ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ❡①♣♦♥❡♥t✐❛❧❧②

❞❡❝r❡❛s✐♥❣✳ ❚❤❡ ❆❇❈s t❤❛t ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ✐♥ ❬✻❪ ❛r❡ ❣✐✈❡♥ ❜②

∂t(∂nu+∂tu) =κ 4 −γ

∂nu−κ 4 +γ

∂tu♦♥Σ. ✭✶✳✶✮

❚❤❡② ❛r❡ s❡❝♦♥❞✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ ❛ ♣❛r❛♠❡t❡r γ✳ ❲❤❡♥ γ = 0✱

✇❡ r❡❝♦✈❡r t❤❡ ❝♦♥❞✐t✐♦♥

∂t(∂nu+∂tu) = κ

4∂nu−κ

4∂tu♦♥Σ

❛♥❞ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤✐s ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝✉r✈❛t✉r❡ ❝♦♥❞✐t✐♦♥

∂nu+∂tu+κ

2u= 0♦♥Σ.

❘❡❣❛r❞✐♥❣ ♥✉♠❡r✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ✇❡ ❤❛✈❡✱ ❛t ❧❡❛st ❢♦r♠❛❧❧② Z

Ω

∂_t²uϕdx+ Z

Ω∇u· ∇ϕdx= Z

Σ

∂nuϕdσ.

❍❡♥❝❡ t❤❡ ♥❡✇ ❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡ ✐♥❝❧✉❞❡❞ ✈❛r✐❛t✐♦♥❛❧② ✐❢ ✐t r❡❛❞s ❛s ∂nu=Bu

♦♥ Σ✇❤❡r❡ B ✐s ❛ ❜♦✉♥❞❛r② ♦♣❡r❛t♦r✳ ■❢ B ✐s ❞✐✛❡r❡♥t✐❛❧✱ t❤❡ s♣❛rs✐t② ♦❢ t❤❡

❞✐s❝r❡t✐③❛t✐♦♥ ♠❛tr✐❝❡s ✐s ♣r❡s❡r✈❡❞✳ ◆♦✇✱ ✐♥❝❧✉❞✐♥❣ ✭✶✳✶✮ ❞✐r❡❝t❧②✱B ✐s ♣s❡✉❞♦✲

❞✐✛❡r❡♥t✐❛❧✳ ❍❡♥❝❡ t❤❡ ❝♦♥❞✐t✐♦♥ ❣❡♥❡r❛t❡s ❛ ❤✐❣❤ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✳ ❚❤❛t

✐s ✇❤② ✇❡ ♣r♦♣♦s❡ t♦ ❞❡✜♥❡ ❛♥ ❛✉①✐❧✐❛r② ✉♥❦♥♦✇♥ t♦ ♦❜t❛✐♥ ❛♥ ❆❇❈ ❡❛s✐❡r t♦

✐♥tr♦❞✉❝❡ ✐♥ t❤❡ ❢♦r♠✉❧❛t✐♦♥✳ ❚❤❡ ❆❇❈ ✭✶✳✶✮ ✐s r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s ✿

∂nu=−∂tu−κ 2

∂t−κ

4 +γ−1

∂tu♦♥Σ

❛♥❞ ✇❡ ❞❡✜♥❡ψ ❛s t❤❡ s✉r❢❛❝❡ ✜❡❧❞ s❛t✐s❢②✐♥❣

∂t−κ 4 +γ

ψ=∂tu♦♥Σ.

❚❤❡♥ t❤❡ s♦❧✉t✐♦♥us❛t✐s✜❡s

∂nu+∂tu+κ

2ψ= 0♦♥Σ,

✇❤✐❝❤ ❝❛♥ ❜❡ ❡❛s✐❧② ✐♥❝❧✉❞❡❞ ✐♥t♦ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✳

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✹

✷ ▼❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s

❲❡ ❝♦♥s✐❞❡r t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ♠✐①❡❞ ♣r♦❜❧❡♠✿ ✜♥❞(u, ∂tu, ψ)s♦❧✉t✐♦♥ t♦



































∂_t²u− △u= 0 ✐♥Ω×(0,+∞) ; u(0, x) =u0(x), ∂tu(0, x) =u1(x) ✐♥Ω;

ψ(0, x) =ψ0(x) ♦♥Σ;

u= 0 ♦♥Γ×(0,+∞) ;

∂nu+∂tu+κ

2ψ= 0 ♦♥Σ×(0,+∞) ;

∂t−κ 4 +γ

ψ=∂tu ♦♥Σ×(0,+∞).

✭✷✳✶✮

❚❤❡ ❢✉♥❝t✐♦♥κ✐s t❤❡ ❝✉r✈❛t✉r❡ ♦❢Σ❛♥❞γ✐s ❛ r❡❣✉❧❛r ♣❛r❛♠❡t❡r ❞❡✜♥❡❞ ♦♥Σ✳

❚❤❡ ❞♦♠❛✐♥ Ω✐s ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ❛♥❞ ✐ts ❜♦✉♥❞❛r②∂Ω = Γ∪Σ✐s ❛ss✉♠❡❞

t♦ ❜❡ r❡❣✉❧❛r✱ ✇✐t❤ Γ∩Σ =∅✭s❡❡ ❋✐❣ ✶✮✳

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡ ♣❛r❛♠❡t❡r ❢✉♥❝t✐♦♥ γ ❝❤❡❝❦s t❤❡ ❢♦❧✲

❋✐❣✉r❡ ✶✿ ❙t✉❞✐❡❞ ❞♦♠❛✐♥

❧♦✇✐♥❣ ❝♦♥❞t✐♦♥

γ(x)> κ(x)

4 ,∀x∈Σ. ✭✷✳✷✮

❲❡ ♣r♦♣♦s❡ t♦ st✉❞② t❤❡ ♣r♦❜❧❡♠ ✭✷✳✶✮ ❜② ✉s✐♥❣ t❤❡ t❤❡♦r② ♦❢ ❍✐❧❧❡✲❨♦s✐❞❛✳ ❲❡

✜rst tr❛♥s❢♦r♠ ✭✷✳✶✮ ✐♥ ❛ ✜rst ♦r❞❡r s②st❡♠ ✐♥ t✐♠❡✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛♥ ❛✉①✐❧✐❛r②

✉♥❦♥♦✇♥v ❞❡✜♥❡❞ ❜②v=∂tu✳ ❚❤❡ ✈❡❝t♦rU = (u, v, ψ)✐s t❤✉s s♦❧✉t✐♦♥ t♦

dU

dt =AU, A=









0 Id 0

△ 0 0

0 1 ^κ₄ −γ







 ✭✷✳✸✮

✇✐t❤ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

u= 0♦♥Γ×(0,+∞) ✭✷✳✹✮

❛♥❞

∂nu+∂tu+κ

2ψ= 0♦♥Σ×(0,+∞). ✭✷✳✺✮

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✺

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ♣r♦❜❧❡♠ ✭✷✳✸✮ ❛♥❞ ✇❡ ✇✐❧❧ ✐♥t❡r❡st

♦✉rs❡❧✈❡s ♦♥ ✐ts s♦❧✉t✐♦♥ ✐♥ s✉✐t❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡s✳

▲❡t ✉s ✜rst ✐♥tr♦❞✉❝❡H ❛s t❤❡ ♣r♦❞✉❝t s♣❛❝❡ ❞❡✜♥❡❞ ❜② H =H_Γ¹(Ω)×L²(Ω)×L²(Σ)

✇❤❡r❡

H_Γ¹(Ω) ={h1∈H¹(Ω), h1= 0♦♥Γ}.

❲❡ ❡q✉✐♣H ✇✐t❤ t❤❡ ❍✐❧❜❡rt✐❛♥ ❣r❛♣❤ ♥♦r♠

k(h1, h2, h3)k^H=

kh1k²L²(Ω)+k∇h1k²L²(Ω)+kh2k²L²(Ω)+kh3k²L²(Σ)

1/2

.

▲❡tV ❜❡ t❤❡ ♣r♦❞✉❝t s♣❛❝❡ ❞❡✜♥❡❞ ❜②

V ={(v1, v2, ϕ)∈H, A(v1, v2, ϕ)∈H, ∂nv1+v2+κ

2ϕ= 0♦♥Σ}.

❚❤❡ s♣❛❝❡ V ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❞♦♠❛✐♥ ♦❢ A✳ ❇② ❡♥❢♦r❝✐♥❣A(v1, v2, ϕ) ∈H✱

✇❡ ✐♠♣r♦✈❡ t❤❡ r❡❣✉❧❛r✐t② ♦❢ ❡❛❝❤ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✉♥❦♥♦✇♥✳ ■♥❞❡❡❞✱ ✇❡ t❤❡♥

❤❛✈❡ v2 ∈ H_Γ¹(Ω) ❛♥❞ △v1 ∈ L²(Ω)✳ ❚❤❡♥ v2 ∈ H_Γ¹(Ω) ✐♠♣❧✐❡s t❤❛t v2|Σ ✐s

❞❡✜♥❡❞ ✐♥H^1/2(Σ)❛♥❞△v1∈L²(Ω)✐♠♣❧✐❡s t❤❛t∂nv1|Σ ∈H^−1/2(Σ)✱ ❦♥♦✇✐♥❣

t❤❛tv1∈H¹(Ω)✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❧❛t✐♦♥∂nv1+v2+^κ₂ϕ= 0♦♥Σ✐♠♣r♦✈❡s t❤❡

r❡❣✉❧❛r✐t② ♦❢∂nv1|Σ s✐♥❝❡v2+^κ₂ϕ∈L²(Σ)✳ ❍❡♥❝❡✱ ❛s ❛ rés✉♠é✱ ✇❡ ❤❛✈❡

V = {(v1, v2, ϕ)∈H,△v1∈L²(Ω), v2∈H_Γ¹(Ω), ∂nv1|Σ ∈L²(Σ),

∂nv1+v2+κ

2ϕ= 0♦♥Σ}.

❋✐rst ♦❢ ❛❧❧ ✇❡ r❡❝❛❧❧ t❤❡ ●r❡❡♥ ❢♦r♠✉❧❛ ✇❡ ✇✐❧❧ ✉s❡✿ ❢♦r ❛❧❧(u, v)∈H¹(Ω)×H¹(Ω) s✉❝❤ t❤❛t△u∈L²(Ω)✱ ✇❡ ❤❛✈❡

Z

Ω△uv dx=− Z

Ω∇u· ∇v dx+< ∂nu, v >_H−1/2(∂Ω),H^1/2(∂Ω). ✭✷✳✻✮

▲❡♠♠❛ ✷✳✶✳ ▲❡tκ❜❡ ❣✐✈❡♥ ✐♥ L^∞(Σ) ❛♥❞ s✉❝❤ t❤❛tmin

x∈Σκ(x) =κ0>0✳ ❚❤❡♥✱

❢♦r ❛❧❧h∈H✱ t❤❡ q✉❛♥t✐t② k|hk|=

Z

Ω|∇h1|²+|h2|²dx+ Z

Σ

κ 2|h3|²dσ

1/2

✐s ❛ ♥♦r♠ ♦♥ H ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♥♦r♠ khk^H✳

Pr♦♦❢✳ ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t k∇ · kL²(Ω) ❞❡✜♥❡s ❛ ♥♦r♠ ♦♥ H_Γ¹(Ω) ✇❤✐❝❤ ✐s

❡q✉✐✈❛❧❡♥t t♦ t❤❡ st❛♥❞❛r❞ ♥♦♠ ✐♥ H¹(Ω)✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ P♦✐♥❝❛ré

✐♥❡q✉❛❧✐t②✳ ❍❡♥❝❡✱ s✐♥❝❡ t❤❡ ❝✉r✈❛t✉r❡ κ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ✐♥ L^∞(Σ) ✇✐t❤

minx∈Σ|κ(x)|>0✱ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t❤❛tk| · k|❞❡✜♥❡s ❛ ♥♦r♠ ♦♥ H ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ♥♦r♠ k · k^H✳

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ❞❡♥♦t❡ ❜②(,)t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ♥♦r♠

k| k|✳

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✻

▲❡♠♠❛ ✷✳✷✳ ▲❡t κ ❛♥❞ γ ❜❡ ❣✐✈❡♥ s✉❝❤ t❤❛t ✭✷✳✷✮ ✐s ❝❤❡❝❦❡❞✳ ❚❤❡♥✱ ❢♦r ❛❧❧

v∈V✱ ✇❡ ❤❛✈❡

(Av, v)≤0.

Pr♦♦❢✳ ▲❡tv= (v1, v2, ϕ)✐♥V✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢A✱Av= v2,△v1, v2+ ^κ₄ −γ ϕ

❚❤❡♥✱ ✇❡ ❤❛✈❡ ✳

(Av, v) = Z

Ω∇v2· ∇v1dx+ Z

Ω△v1v2dx+ Z

Σ

κ 2

v2+κ 4 −γ

ϕ ϕ dσ.

❯s✐♥❣ t❤❡ ●r❡❡♥ ❢♦r♠✉❧❛ ✭✷✳✻✮✱ ✇❡ ❣❡t (Av, v) =

Z

Ω∇v2· ∇v1dx − Z

Ω∇v2· ∇v1dx + Z

Σ

∂nv1v2dσ+ Z

Σ

κ 2v2ϕ dσ +< ∂nv1, v2>_H−1/2(Γ),H^1/2(Γ)+

Z

Σ

κ 2

κ 4 −γ

|ϕ|²dσ.

▼♦r❡♦✈❡r✱ ✐♥V✱ ✇❡ ❤❛✈❡v1|Γ =v2|Γ= 0❛♥❞ ♦♥Σ✱∂nv1=−v2−^κ2ϕ.❍❡♥❝❡✱

(Av, v) =− Z

Σ|v2|²dσ− Z

Σ

κ

2ϕv2dσ+ Z

Σ

κ

2ϕv2dσ+ Z

Σ

κ 2

κ 4 −γ

|ϕ|²dσ.

◆♦✇✱ s✐♥❝❡ ^κ₂ ^κ₄ −γ

≤0 ♦♥Σ✱ ✇❡ ❣❡t t❤❛t ❢♦r ❛❧❧v ✐♥V (Av, v) =−

Z

Σ|v2|²dσ+ Z

Σ

κ 2

κ 4 −γ

|ϕ|²dσ≤0,

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✷✳✷✳

▲❡♠♠❛ ✷✳✸✳ ❚❤❡ ♦♣❡r❛t♦r A✱ ✇✐t❤ ❞♦♠❛✐♥ V✱ ✐s ♠❛①✐♠❛❧✳

Pr♦♦❢✳ ●✐✈❡♥ f = (f1, f2, f3) ✐♥ H✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♠✐①❡❞ ♣r♦❜❧❡♠✿

✜♥❞v∈V s✉❝❤ t❤❛t(A−I)v=f✳

❲❡ t❤✉s s❡❡❦v= (v1, v2, ϕ)∈V s✉❝❤ t❤❛t



























v2−v1=f1 ✐♥Ω;

△v1−v2=f2 ✐♥ Ω;

v2+ ^κ₄ −γ−1

ϕ=f3 ♦♥Σ;

v1= 0♦♥Γ;

∂nv1+v2+^κ₂ϕ= 0♦♥Σ.

✭✷✳✼✮

❋✐rst ♦❢ ❛❧❧✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✭✷✳✼✮ ❤❛s ❛ s♦❧✉t✐♦♥ ✐♥ V✳ ❚❤❡♥✱ ❜② r❡♠♦✈✐♥❣v2 t❤❛♥❦s t♦ t❤❡ ❡q✉❛t✐♦♥

v2=f1+v1✐♥ Ω,

❛♥❞ϕt❤❛♥❦s t♦ t❤❡ t❤✐r❞ ❡q✉❛t✐♦♥

ϕ= f3−f1−v1 κ

4 −γ−1 . ✭✷✳✽✮

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✼

✇❡ ♦❜t❛✐♥ t❤❛tv1 ✐s s♦❧✉t✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠











− △v1+v1= ˜f ✐♥ Ω;

v1= 0♦♥Γ;

∂nv1+αv1= ˜g♦♥Σ.

✭✷✳✾✮

✇✐t❤ f˜:=−(f2+f1) ✐♥L²(Ω),

˜

g:=− 1 + κ 2 1 +γ−^κ4

!

f1+ κ

2 1 +γ−^κ4

f3✐♥ L²(Σ),

❛♥❞

α= 1 + κ 2 1 +γ−^κ4

>0.

■t ✐s ♦❜✈✐♦✉s t❤❛t✱ s✐♥❝❡ κ∈L^∞(Σ) ❛♥❞α >0 ❜② ❤②♣♦t❤❡s✐s✱ t❤❡ ❢✉♥❝t✐♦♥❛❧

|v|^1,α=

kvk²H¹(Ω)+ Z

Σ

α|v|²dσ 1/2

❞❡✜♥❡s ❛ ♥♦r♠ ✐♥H¹(Ω)✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ st❛♥❞❛r❞ ♥♦r♠k k^H1(Ω)✳ ▲❡t T Ω

❜❡ t❤❡ s♣❛❝❡ ♦❢ t❡st ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❜② T Ω

={φ∈ D Ω

, φ|Γ= 0}

■t ✐s ❞❡♥s❡ ✐♥H_Γ¹(Ω)❛♥❞ ✐❢ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✭✷✳✾✮ ❤❛s ❛ s♦❧✉t✐♦♥✱ ✇❡

❤❛✈❡

∀φ∈ T Ω ,−

Z

Ω△v1φ dx+ Z

Ω

v1φ dx= Z

Ω

f φ dx.˜

❇② ✉s✐♥❣ t❤❡ ●r❡❡♥ ❢♦r♠✉❧❛ ✭✷✳✻✮✱ ✇❡ ❣❡t

∀φ∈ T Ω ,

Z

Ω∇v1∇φ dx−< ∂nv1, φ >_H−1/2(∂Ω),H^1/2(∂Ω) + Z

Ω

v1φ dx= Z

Ω

f φ dx.˜

▼♦r❡♦✈❡r✱φ|Γ= 0✳ ❚❤❡r❡❢♦r❡✱

∀φ∈ T Ω ,

Z

Ω∇v1∇φ dx− Z

Σ

∂nv1φ dσ+ Z

Ω

v1φ dx= Z

Ω

f φ dx.˜

❚❤❡♥ ❜② ✉s✐♥❣ t❤❛t∂nv1= ˜g−αv1 ♦♥Σ✱ ✇❡ ♦❜t❛✐♥

∀φ∈ T Ω ,

Z

Ω∇v1∇φ dx + Z

Σ

αv1φ dσ+ Z

Ω

v1φ dx= Z

Ω

f φ dx˜ + Z

Σ

˜ gφ dσ.

✭✷✳✶✵✮

▲❡ta(·,·)t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ ❞❡✜♥❡❞ ❜② a(v1, φ) =

Z

Ω∇v1∇φ dx+ Z

Σ

αv1φ dσ+ Z

Ω

v1φ dx.

■t ✐s ♦❜✈✐♦✉s t❤❛t a(·,·) ✐s ❝♦♥t✐♥✉♦✉s ♦♥H_Γ¹(Ω)×H_Γ¹(Ω) ❛♥❞ H_Γ¹(Ω)✲❝♦❡r❝✐✈❡✱

s✐♥❝❡a(v1, φ)❝♦rr❡s♣♦♥❞s ❡①❛❝t❧② t♦ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ❞❡✜♥❡❞ ❢r♦♠ t❤❡ ♥♦r♠

| · |^1,α✳ ▲❡tl(·)❜❡ t❤❡ ❧✐♥❡❛r ❢♦r♠ ❞❡✜♥❡❞ ❜② l(φ) =

Z

Ω

f φ dx˜ + Z

Σ

˜ gφ dσ.

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✽

❙✐♥❝❡ t❤❡ ♣❛✐r ( ˜f ,g)˜ ❜❡❧♦♥❣s t♦L²(Ω)×L²(Σ)✱l(·)✐s ❝♦♥t✐♥✉♦✉s ✐♥H_Γ¹(Ω)✳

❚❤❡♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢❛❝t t❤❛tT Ω

✐s ❞❡♥s❡ ✐♥H_Γ¹(Ω)✱ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ✭✷✳✶✵✮

❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ H_Γ¹(Ω)✿

∀φ∈H_Γ¹(Ω), a(v1, φ) =l(φ)

❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ ▲❛①✲▼✐❧❣r❛♠ t❤❡♦r❡♠✱ t❤❡ ♣r♦❜❧❡♠

∀φ∈H_Γ¹(Ω), a(v1, φ) =l(φ)

❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥v1✐♥ H_Γ¹(Ω)✳ ■♥ ♣❛rt✐❝✉❧❛r✱

∀φ∈ D(Ω)⊂H_Γ¹(Ω), Z

Ω∇v1∇φ dx+ Z

Ω

v1φ dx= Z

Ω

f φ dx.˜

❲❡ t❤❡♥ ❞❡❞✉❝❡ t❤❛t

∀φ∈ D(Ω), < v1− △v1−f , φ >= 0,˜

✇❤✐❝❤ ♠❡❛♥s t❤❛t

v1− △v1= ˜f ✐♥D^′(Ω).

❚❤✐s ✐❞❡♥t✐t② ❛❧❧♦✇s ✉s t♦ ❣✐✈❡ ❛ s❡♥s❡ t♦ △v1 ✐♥ L²(Ω)✳ ❚❤❡r❡❢♦r❡✱ ∂nv1|∂Ω∈ H^−1/2(∂Ω)❛♥❞ ✇❡ ❛❧s♦ ❤❛✈❡

∀φ∈ T Ω ,

Z

Ω∇v1∇φ dx+ Z

Σ

αv1φ dσ+ Z

Ω

v1φ dx= Z

Ω

f φ dx˜ + Z

Σ

˜ gφ dσ.

❯s✐♥❣ t❤❡ ●r❡❡♥ ❢♦r♠✉❧❛ ✭✷✳✻✮✱ ✇❡ ❣❡t ∀φ∈ T Ω , Z

Ω△v1φ dx+< ∂nv1, φ >_H−1/2(∂Ω),H^1/2(∂Ω)+ Z

Σ

αv1φ dσ+ Z

Ω

v1φ dx= Z

Ω

f φ dx˜ + Z

Σ

˜ gφ dσ,

✐✳❡✳

∀φ∈ T Ω

,+< ∂nv1, φ >_H−1/2(∂Ω),H^1/2(∂Ω)+ Z

Σ

αv1φ dσ= Z

Σ

˜ gφ dσ.

◆♦✇✱ ✇❡ ❤❛✈❡φ|Γ= 0 ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t

< ∂nv1+ 1 + κ 2 1 +γ−^κ4

!

v1−g, φ >˜ _H−1/2(Σ),H^1/2(Σ)= 0,

❛♥❞ ✇❡ t❤❡♥ ❤❛✈❡

∂nv1+ 1 + κ 2 1 +γ−^κ4

!

v1= ˜g ♦♥Σ.

❚❤❡ ❡①✐st❡♥❝❡ ♦❢ v1s♦❧✉t✐♦♥ t♦ ✭✷✳✾✮ ✐s t❤✉s ♣r♦✈❡❞✳

❙✐♥❝❡ v1 ❛♥❞ f1 ❛r❡ ✐♥ H¹(Ω)✱ ✇❡ ❞❡❞✉❝❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ v2 = f1 +v1 ✐♥

H¹(Ω)✳ ▼♦r❡♦✈❡r s✐♥❝❡ v2|Σ ❛♥❞ f3 ❛r❡ ✐♥ L²(Σ)✱ ✇❡ ❞❡❞✉❝❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■ ✾

ϕ= f3−f1−v1 κ

4−γ−1 ✐♥L²(Σ)✳

❚♦ ❝♦♠♣❧❡t❡ t❤❡ ♣r♦♦❢✱ ✇❡ ❤❛✈❡ t♦ ❝❤❡❝❦ t❤❛t ❛❝t✉❛❧❧②(v1, v2, ϕ)∈V✱ ✇❤✐❝❤ ✐s

♦❜✈✐♦✉s ❢r♦♠

v2=f1+v1 ✐♥Ω αϕ=f3−v2♦♥Σ

❛♥❞f1∈H_Γ¹(Ω)✱f3∈L²(Σ)✳

❆s ❛ ❝♦♥❝❧✉s✐♦♥✱

❚❤❡♦r❡♠ ✷✳✹✳ ▲❡t (u0, u1, ϕ0)✐♥V✳ ❚❤❡ ♣r♦❜❧❡♠ ✭✷✳✶✮ ❛❞♠✐ts ❛ ✉♥✐q✉❡ s♦❧✉✲

t✐♦♥us✉❝❤ t❤❛t

(u, ∂tu, ϕ)∈C¹([0,+∞[;V)∩C⁰([0,+∞[;H). ✭✷✳✶✶✮

Pr♦♦❢✳ ❚❤❡ t✇♦ ♣r❡✈✐♦✉s ❧❡♠♠❛s s❤♦✇ t❤❛t t❤❡ ♦♣❡r❛t♦r A ✐s ❛ ♠❛①✐♠❛❧ ❞✐s✲

s✐♣❛t✐✈❡ ♦♣❡r❛t♦r ✐♥ ✐ts ❞♦♠❛✐♥V✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❍✐❧❧❡✲❨♦s✐❞❛ t❤❡♦r❡♠ ❬✶✷❪

t❤❡ ♣r♦❜❧❡♠ ✭✷✳✶✮ ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ s♦❧✉t✐♦♥ U = (u, v, ψ)s✉❝❤ t❤❛t (u, v, ψ)∈C¹([0,+∞[;V)∩C⁰([0,+∞[;H). ✭✷✳✶✷✮

A✐s t❤✉s t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ ❛ s❡♠✐✲❣r♦✉♣ ♦❢ ❝♦♥tr❛❝t✐♦♥Z(t)❛♥❞ ✇❡

❝❛♥ ❞❡✜♥❡ t❤❡ ✜♥✐t❡ ❡♥❡r❣② s♦❧✉t✐♦♥ ♦❢ ✭✷✳✶✮ ✇✐t❤ ✐♥✐t✐❛❧ ❞❛t❛(u0, u1, ψ0) ✐♥ V

✐♥ s✉❝❤ ❛ ✇❛② t❤❛t

(u, v, ψ) =Z(t)(u0, u1, ψ0)∈C¹([0,+∞[;V)∩C⁰([0,+∞[;H) ;

✇❤✐❝❤ ❡♥❞s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠✳

✸ ▲♦♥❣ t✐♠❡ ❜❡❤❛✈✐♦r

❚❤❡ r❡s✉❧ts ♦❢ ❙❡❝t✐♦♥ ✷ ❝❛♥ ❜❡ ❡♥r✐❝❤❡❞ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❞❡✜♥❡❞

♦♥H ❜②

E(h1, h2, h3) = 1 2

Z

Ω |∇h1|²+|h2|² dx+1

2 Z

Σ

κ

2|h3|²dσ.

■❢κ(x)>0❢♦r ❛❧❧x∈Σ✱E❞❡✜♥❡s ❛♥ ❡♥❡r❣② ♦♥H ❛♥❞E^1/2✐s ♦❜✈✐♦✉s❧② ❛ ♥♦r♠

✐♥ H ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♥♦r♠k k✱ ❛❝❝♦r❞✐♥❣ t♦ ▲❡♠♠❛ ✷✳✶✳ ❚❤❡♥✱ E(u, ∂tu, ψ)

❞❡✜♥❡s ❛♥ ❡♥❡r❣② ♦♥ H✳

▼♦r❡♦✈❡r✱

▲❡♠♠❛ ✸✳✶✳ ❋♦r ❛❧❧(u0, u1, ψ0)∈V, t7−→ E(u, ∂tu, ψ)✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ✐s

❞❡❝r❡❛s✐♥❣ ✉♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ ✭✷✳✷✮✿

γ(x)> κ(x)

4 ,∀x∈Σ.

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■✶✵

Pr♦♦❢✳ ■♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t ✐❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s(u0, u1, ψ0)

❛r❡ ✐♥ V✱E(u, ∂tu, ψ)∈C¹([0,+∞[)✳ ▼♦r❡♦✈❡r✱

d

dtE(u, ∂tu, ψ) = Z

Ω∇(∂tu)· ∇u dx+ Z

Ω

∂tu∂_t²u dx+ Z

Σ

κ

2ψ∂tψ dσ. ✭✸✳✶✮

❯s✐♥❣ t❤❡ ●r❡❡♥ ❢♦r♠✉❧❛ ✭✷✳✻✮ ❛♥❞ t❤❡ r❡❧❛t✐♦♥∂_t²u=△u✐♥Ω✱ ✇❡ ♦❜t❛✐♥

d

dtE(u, ∂tu, ψ) =− Z

Ω

∂tu△u dx+

Z

∂Ω

∂nu∂tu dσ+ Z

Ω

∂tu△u dx+ Z

Σ

κ

2ψ∂tψ dσ.

❙✐♥❝❡∂tu|Γ = 0❛♥❞∂nu=−∂tu−^κ2ψ♦♥Σ✱

d

dtE(u, ∂tu, ψ) =− Z

Σ|∂tu|²dσ− Z

Σ

κ

2ψ∂tudσ.

❆t ❧❛st✱ s✐♥❝❡∂tψ=∂tu− γ−^κ4

ψ✱ ✇❡ ❣❡t d

dtE(u, ∂tu, ψ) =− Z

Σ

κ 2

γ−κ 4

|ψ|²dσ+ Z

Σ|∂tu|²dσ

.

✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t

d

dtE(u, ∂tu, ψ)≤0,

❛♥❞ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✸✳✶✳

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ψ0 = 0 ♦♥ Σ✳ ❚❤✐s ✐s ❛ ♥❡❝❡ss❛r②

❝♦♥❞✐t✐♦♥ ❢♦r ✭✷✳✶✮ t♦ ❜❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ✐♥✐t✐❛❧ ♣r♦❜❧❡♠

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∂_t²u− △u= 0✐♥Ω×(0,+∞) ;

u(0, x) =u0(x), ∂tu(0, x) =u1(x)✐♥Ω;

u= 0♦r∂nu= 0♦♥Γ×(0,+∞) ;

∂t(∂nu+∂tu) =κ 4 −γ

∂nu−κ 4 +γ

∂tu♦♥Σ×(0,+∞).

❚❤❡♦r❡♠ ✸✳✷✳ ❯♥❞❡r ✭✷✳✷✮✱ ❢♦r ❛❧❧(u0, u1,0)∈V✱

t→+∞lim E(u, ∂tu, ψ) = 0.

Pr♦♦❢✳ ❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❛tA✐s t❤❡ ❣❡♥❡r❛t♦r ♦❢ ❛ ❝♦♥t✐♥✉♦✉s ❝♦♥tr❛❝t✐♦♥

s❡♠✐✲❣r♦✉♣Z(t)✳ ❆s ✐t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ♦♥ ❛ ❞❡♥s❡ s✉❜s♣❛❝❡ ♦❢

V =D(A)✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ✐♥✐t✐❛❧ ❞❛t❛(u0, u1,0)✐♥D(A²)✱ ✇❤❡r❡

D(A²) ={(v1, v2, ψ)∈V, A(v1, v2, ψ)∈V}

✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠ ❣r❛♣❤

k(v1, v2, ψ)k^D(A2)=k(v1, v2, ψ)k^V +kA(v1, v2, ψ)k^V +kA²(v1, v2, ψ)k^V.

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■✶✶

❋♦r ❛♥② s♦❧✉t✐♦♥ t♦ ✭✷✳✶✮✱ ✇❡ ❤❛✈❡

k(u, ∂tu, ψ)k^D(A2) =kZ(t)(u0, u1,0)k^D(A2)

=kZ(t)(u0, u1,0)k^V +kA(Z(t)(u0, u1,0))k^V +kA²(Z(t)(u0, u1,0))k^V.

❆sA✱ A² ❛♥❞Z(t)❛r❡ ❝♦♠♠✉t✐♥❣ ♦♥D(A²)✱

k(u, ∂tu, ψ)kD(A²) =kZ(t)(u0, u1,0)k^V +kZ(t)A(u0, u1,0)k^V +kZ(t)A²(u0, u1,0)k^V.

❙✐♥❝❡ Z(t) ✐s ❝♦♥t✐♥✉♦✉s ✐♥ V✱ ✇❡ ❞❡❞✉❝❡ ❡❛s✐❧② t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡

❝♦♥st❛♥tC s✉❝❤ t❤❛t

k(u, ∂tu, ψ)k^D(A2)≤Ck(u0, u1,0)k^D(A2).

❲❡ t❤✉s ❤❛✈❡ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ✐♥ D(A²)✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t ✇❡

❝❛♥ ❡①tr❛❝t ❛ s✉❜s❡q✉❡♥❝❡ ❞❡♥♦t❡❞ ❜② Z(tk)(u0, u1,0)✇❤✐❝❤ ✇❡❛❦❧② ❝♦♥✈❡r❣❡s t♦ (u∞, v∞, ψ∞) ✐♥ D(A²)✳ ◆♦✇✱ ❧❡t ✉s ❞❡♥♦t❡ ❜② (u(tk), v(tk)) t❤❡ s❡q✉❡♥❝❡

t❤❛t ✐s ❝♦♥✈❡r❣✐♥❣ t♦(u∞, v∞)✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢D(A²)✱(u(tk), v(tk))✐s ❜♦✉♥❞❡❞

✐♥ H^3/2(Ω)×H^3/2(Ω) ❛♥❞△u(tk) ✐s ❜♦✉♥❞❡❞ ✐♥H¹(Ω)✳ ■♥❞❡❡❞✱ ❛♥②(u, v)✐♥

D(A²)s❛t✐s✜❡s

( u∈H¹(Ω),△u∈H¹(Ω), u= 0 ♦♥Γ, ∂nu|Σ ∈L²(Σ) v∈H¹(Ω),△v∈H¹(Ω), v= 0♦♥Γ, ∂nv_|Σ ∈L²(Σ).

❲❡ ❝❛♥ t❤✉s ❞❡❞✉❝❡ t❤❛t(u(tk), v(tk))str♦♥❣❧② ❝♦♥✈❡r❣❡s t♦(u∞, v∞)✐♥H¹(Ω)× H¹(Ω)✳ ▼♦r❡♦✈❡r✱ s✐♥❝❡ △u(tk) str♦♥❣❧② ❝♦♥✈❡r❣❡s ✐♥L²(Ω) ❛♥❞ △u(tk)❝♦♥✲

✈❡r❣❡s t♦△u∞ ✐♥D^′(Ω)✱△u(tk)str♦♥❣❧② ❝♦♥✈❡r❣❡s t♦△u∞✐♥L²(Ω)s✐♥❝❡ t❤❡

❧✐♠✐t ✐s ✉♥✐q✉❡✳

❲❡ t❤❡♥ ❤❛✈❡ t❤❛t∂nu(tk)_|Σ str♦♥❣❧② ❝♦♥✈❡r❣❡s t♦∂nu∞|Σ✐♥H^−1/2(Σ)❛♥❞ t❤❛t v(tk)_|Σ str♦♥❣❧② ❝♦♥✈❡r❣❡s t♦v∞|Σ ✐♥H^1/2(Σ)✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛tφ(tk)str♦♥❣❧②

❝♦♥✈❡r❣❡s t♦ φ∞ ✐♥ H^−1/2(Σ)✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤✐s r❡s✉❧t ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐s ♥♦t s✉✣❝✐❡♥t t♦ ❤❛✈❡ t❤❛t (u(tk), v(tk), φ(tk))str♦♥❣❧② ❝♦♥✈❡r❣❡s t♦ (u∞, v∞, ψ∞)

✐♥ V✳ ❚❤❛t ✐s ✇❤② ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ t❤❛t ❞❡✜♥❡sφ(tk)t♦ s❤♦✇ t❤❛t ✐♥

❢❛❝t✱φ(tk)str♦♥❣❧② ❝♦♥✈❡r❣❡s ✐♥L²(Σ)✳ ❇② ❝♦♥str✉❝t✐♦♥✱φ✐s s♦❧✉t✐♦♥ t♦✿

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∂tφ+ γ−κ

4

φ=∂tu ♦♥Σ×(0,+∞)

φ(0, x) = 0 ♦♥Σ

❛♥❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❉✉❤❛♠❡❧ ❢♦r♠✉❧❛✱ ✇❡ ❤❛✈❡

φ(t, x) = Z t

0

e^s−tv(s, x)ds.

❲❡ ❦♥♦✇ t❤❛tv✐s ❜♦✉♥❞❡❞ ✐♥H^1/2(Σ)✳ ❍❡♥❝❡ ✇❡ ❤❛✈❡✿ ❢♦r ❛♥②ξ∈H^−1/2(Σ)✱

< φ, ξ >_H^1/2_,H−1/2 =<

Z t 0

e^s−tvds, ξ >_H^1/2_,H−1/2

= Z t

0

e^s−t< v, ξ >H^1/2,H^−1/2 ds

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■✶✷

❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢❛❝t t❤❛t v(s) ✐s ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ✇✐t❤ r❡s♣❡❝t t♦ s✳ ❚❤✉s

✇❡ ♦❜t❛✐♥

|< φ, ξ >_H^1/2_,H−1/2 | ≤ kvkH^1/2(Σ)kξkH^−1/2(Σ)(1−e^−t)

✇❤✐❝❤ ✐♠♣❧✐❡s

kφkH^1/2(Σ)≤(1−e^−t)kvkH^1/2(Σ).

❲❡ t❤✉s ❤❛✈❡ ♣r♦✈❡❞ t❤❛t φ(t) ✐s ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ✐♥ H^1/2(Σ) ❛♥❞ ✇❡ ❝❛♥

t❤❡♥ ❞❡❞✉❝❡ t❤❛tφ(tk)str♦♥❣❧② ❝♦♥✈❡r❣❡s t♦φ∞✐♥ L²(Σ)✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢

t❤❡ ❝♦♠♣❛❝t ✐♥❥❡❝t✐♦♥ ❢r♦♠H^1/2(Σ) ✐♥t♦L²(Σ)✳

❆s ❛ ❝♦♥❝❧✉s✐♦♥✱(u(tk), v(tk), φ(tk))str♦♥❣❧② ❝♦♥✈❡r❣❡s t♦(u∞, v∞, ψ∞)✐♥V✳

❙✐♥❝❡t7−→ E(u, ∂tu)✐s ❝♦♥t✐♥✉♦✉s✱ ✇❡ t❤✉s ❤❛✈❡

t→+∞lim E(u, ∂tu, ψ) = lim

t→+∞E(Z(t)(u0, u1,0))

= lim

tk→+∞E(Z(tk)(u0, u1,0))

=E(u∞, v∞, ψ∞).

❲❡ ❛❧s♦ ❤❛✈❡✱ ❢♦r ❛❧❧s♣♦s✐t✐✈❡✱

t→+∞lim E(Z(t+s)(u0, u1,0)) = lim

tk→+∞E(Z(s)Z(tk)(u0, u1,0)) =E(Z(s)(u∞, v∞, ψ∞)).

❚❤❡♥ ✐❢ (w, ∂tw, ϕ) =Z(t)(u∞, v∞, ψ∞)❞❡♥♦t❡s t❤❡ s♦❧✉t✐♦♥ t♦ ♣r♦❜❧❡♠ ✭✷✳✶✮✱

✇✐t❤ ✐♥✐t✐❛❧ ❞❛t❛(u∞, v∞, ψ∞)✐♥D(A)✱ ✇❡ ❤❛✈❡

E(w, ∂tw), ϕ=E(u∞, v∞, ψ∞)❢♦r ❛❧❧t ♣♦s✐t✐✈❡.

❍❡♥❝❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✸✳✶✱ s✐♥❝❡

d

dtE(w, ∂tw, ϕ) =− Z

Σ

κ 2

γ−κ 4

|ϕ|²dσ+ Z

Σ|∂tw|²dσ

,

✇❡ ♥❡❝❡ss❛r✐❧② ❤❛✈❡∂tw= 0♦♥Σ❛♥❞ ϕ= 0 ♦♥Σ✳ ❲❡ t❤❡♥ ❞❡❞✉❝❡ t❤❛tw ✐s s♦❧✉t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠

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∂²_tw− △w= 0 ✐♥Ω×[0,+∞[ w(x,0) =u∞, ∂tw(x,0) =v∞ ✐♥Ω

w= 0 ♦♥Γ×[0,+∞[

∂nw= 0 ♦♥Σ×(0,+∞)

❛♥❞z:=∂tw✐s s♦❧✉t✐♦♥ t♦

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∂_t²z− △z= 0 ✐♥ Ω×[0,+∞[ z(x,0) =v∞, ∂tz(x,0) =△u∞ ✐♥ Ω

z= 0 ♦♥Γ×[0,+∞[

∂nz=z= 0 ♦♥Σ×(0,+∞).

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■✶✸

❙✐♥❝❡∂nz=z= 0♦♥Σ✱ ✇❡ ❞❡❞✉❝❡ t❤❛tz= 0✐♥Ω×[0,+∞[✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡

♦❢ t❤❡ ❍♦❧♠❣r❡♥ t❤❡♦r❡♠ ✭s❡❡ ▲✐♦♥s ❬✶✹❪✮✳ ❚❤❡r❡❢♦r❡✱w ✐s s♦❧✉t✐♦♥ t♦

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△w= 0 ✐♥Ω×[0,+∞[ w(x,0) =u∞ ✐♥Ω

w= 0 ♦♥Γ×[0,+∞[

∂nw= 0 ♦♥Σ×(0,+∞)

✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛tw= 0✐♥Ω×[0,+∞[s✐♥❝❡Ω✐s ❝♦♥♥❡❝t❡❞✳❚❤❡ ♣❛✐r(u∞, v∞)

✐s t❤✉s ❡q✉❛❧ t♦ ③❡r♦✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❛❧s♦ ❣❡t t❤❛t ψ∞ ✐s ❛❧s♦ ❡q✉❛❧ t♦ ③❡r♦

❛♥❞ ❛ ❢♦rt✐♦r✐✱ ✇❡ ❤❛✈❡E(u∞, v∞, ψ∞) = 0✳

❲❡ ❛r❡ ♥♦✇ ✇✐❧❧✐♥❣ t♦ ♣r♦♣♦s❡

❚❤❡♦r❡♠ ✸✳✸✳ ▲❡t(u0, u1,0)✐♥ V✳ ❚❤❡♥ t❤❡ s♦❧✉t✐♦♥ut♦ ✭✷✳✶✮ s❛t✐s✜❡s

t→+∞lim (u, ∂tu, ψ) = (0,0,0)✐♥H.

Pr♦♦❢✳ ■❢ t❤❡ ♣❛✐r(u0, u1,0)✐s ✐♥ V✱ ✇❡ ❦♥♦✇ t❤❛t ✭❝❢✳ ❚❤❡♦r❡♠ ✸✳✷✮

t→+∞lim E(u, ∂tu, ψ) = 0

❛♥❞ t❤❛t E ✐s ❛ ♥♦r♠ ♦♥H✱ ❡q✉✐✈❛❧❡♥t t♦k k✳

❚❤❡r❡❢♦r❡✱

t→+∞lim (u, ∂tu, ψ) = (0,0,0)✐♥ H.

✹ ❊①♣♦♥❡♥t✐❛❧ ❉❡❝❛②

❚❤❡ ❛✐♠ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ st✉❞② t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❡♥❡r❣② ❞❡❝❛② ♦❢ t❤❡ s♦❧✉t✐♦♥

♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✷✳✶✮ ✇❤❡♥ t❤❡ ♦❜st❛❝❧❡ ✐s ❛ s♦✉♥❞✲s♦❢t s❝❛tt❡r❡r ✭u= 0 ♦♥Γ✮✳

❲❡ ✇❛♥t t♦ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts t✇♦ ♣♦s✐t✐✈❡ ❝♦♥st❛♥tC ❛♥❞βs✉❝❤ t❤❛t ❢♦r

❛❧❧ ✐♥✐t✐❛❧ ❞❛t❛✱

E(u, ∂tu, ψ)≤Ce^−βtE(u, ∂tu, ψ)|t=0

✇❤❡r❡

E(u, ∂tu, ψ) = 1 2

Z

Ω |∂tu|²+|∇u|² dx+1

2 Z

Σ

κ

2|ψ|²dσ. ✭✹✳✶✮

❚♦ ♦❜t❛✐♥ s✉❝❤ ❛♥ ✐♥❡q✉❛❧✐t②✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ●r♦♥✇❛❧❧✬s ❧❡♠♠❛ ❛♥❞ ✐t ✐s t❤✉s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥tC s✉❝❤ t❤❛t

Z T

S E(u, ∂tu, ψ)dt≤CE(u, ∂tu, ψ)|t=S ✭✹✳✷✮

✇✐t❤0≤S < T <+∞✳

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■✶✹

✹✳✶ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts

▲❡♠♠❛ ✹✳✶✳ ▲❡t(u0, u1, ψ0)∈V✳ ❚❤❡♥✱

E(T)− E(S) =− Z T

S

Z

Σ|∂tu|²dσdt+ Z T

S

Z

Σ

κ 2

κ 4 −γ

|ψ|²dσdt, ✭✹✳✸✮

✇✐t❤

E(t) =E(u, ∂tu, ψ)(t).

Pr♦♦❢✳ ❲❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❛tE ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞

dE

dt(u, ∂tu, ψ) =− Z

Σ|∂tu|²dσ+ Z

Σ

κ 2

κ 4 −γ

|ψ|²dσ.

❚❤❡♥ ❛❢t❡r ✐♥t❡❣r❛t✐♥❣ ♦♥[S, T]✱ ✇❡ ❣❡t E(T)− E(S) =−

Z T S

Z

Σ|∂tu|²dσdt+ Z T

S

Z

Σ

κ 2

κ 4 −γ

|ψ|²dσdt.

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ❛ss✉♠❡ t❤❛t γ(x)>κ(x)

4 ,∀x∈Σ ✭✹✳✹✮

❛♥❞ t❤❛t Σ✐s str✐❝t❧② ❝♦♥✈❡① s♦ t❤❛t

κ(x)>0, ∀x∈Σ. ✭✹✳✺✮

▲❡♠♠❛ ✹✳✷✳ ❲❡ ❤❛✈❡









 Z T

S

Z

Σ|∂tu|²dσ dt≤ E(S) Z T

S

Z

Σ

κ 2

γ−κ 4

|ψ|²dσ dt≤ E(S)

✭✹✳✻✮

❛♥❞ ✐❢ α_♠✐♥= min

x∈Σ γ−^κ4

✱ ✇❡ ❤❛✈❡

Z T S

Z

Σ

κ

2|ψ|²dσ dt≤ 1

α_♠✐♥E(S) ✭✹✳✼✮

Pr♦♦❢✳ ❲❡ ❦♥♦✇ t❤❛t E(T)− E(S) =−

Z T S

Z

Σ|∂tu|²dσdt+ Z T

S

Z

Σ

κ 2

κ 4 −γ

|ψ|²dσdt

✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦

E(S) = Z T

S

Z

Σ|∂tu|²dσdt+ Z T

S

Z

Σ

κ 2

γ−κ 4

|ψ|²dσdt+E(T).

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■✶✺

❙✐♥❝❡ ❡❛❝❤ t❡r♠ ✐s ♣♦s✐t✐✈❡ ✇❡ ❣❡t









 Z T

S

Z

Σ|∂tu|²dσ dt≤ E(S);

Z T S

Z

Σ

κ 2

γ−κ 4

|ψ|²dσ dt≤ E(S).

❋r♦♠Z T S

Z

Σ

κ 2

γ−κ 4

|ψ|²dσ dt≤ E(S)✱ ✇❡ ♦❜✈✐♦✉s❧② ❣❡t

Z T S

Z

Σ

κ

2|ψ|²dσ dt≤ 1 α_♠✐♥E(S)

✇❤✐❝❤ ❡♥❞s t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✹✳✻✳

▲❡♠♠❛ ✹✳✸✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C >0 s✉❝❤ t❤❛t✱ ❢♦r ❛♥② t≤S≤0✱

Z

Σ|u(t, x)|²dσ≤CE(S). ✭✹✳✽✮

Pr♦♦❢✳ ❚❤❡ tr❛❝❡ ♠❛♣ ❢r♦♠ H¹(Ω)✐♥t♦ H^1/2(Σ)✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ✇❡ ❤❛✈❡

kuk²L²(Σ)≤Ckuk²H^1/2(Σ)≤Ckuk²H¹(Ω).

▼♦r❡♦✈❡r✱us❛t✐s✜❡s t❤❡ P♦✐♥❝❛ré ✐♥❡q✉❛❧✐t② ✇❤✐❝❤ ✐s

❚❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t Cs✉❝❤ t❤❛t

kuk^L2(Ω)≤Ck∇uk^L2(Ω). ✭✹✳✾✮

❚❤✐s ✐♠♣❧✐❡s t❤❛t

kuk²L²(Σ)≤CE(t).

❲❡ ❝♦♥❝❧✉❞❡ ❡❛s✐❧② s✐♥❝❡t7−→ E(t)✐s ❞❡❝r❡❛s✐♥❣✳

▲❡tm(x)❜❡ ❛ ❢✉♥❝t✐♦♥ ✐♥C¹( ¯Ω)³✳

▲❡♠♠❛ ✹✳✹✳ ❲❡ ❤❛✈❡

Z

Ω

∂tu(m· ∇u)dx T

S

−1 2

Z T S

Z

∂Ω

(m·n)|∂tu|²dσ dt+1 2

Z T S

Z

Ω

❞✐✈ m|∂tu|²dx dt+

Z T S

Z

Ω∇u· ∇(m· ∇u)dx dt− Z T

S

Z

∂Ω

∂nu(m· ∇u)dσ dt= 0.

✭✹✳✶✵✮

Pr♦♦❢✳ ❚❤✐s ✐❞❡♥t✐t② ❤❛s ❜❡❡♥ ✉s❡❞ ✐♥ s❡✈❡r❛❧ ♣❛♣❡rs✳ ❲❡ r❡❢❡r ❢♦r ✐♥st❛♥❝❡ t♦

❬✶✸❪✳ ◆❡✈❡rt❤❡❧❡ss✱ ✇❡ r❡❝❛❧❧ ✐ts ❝♦♥str✉❝t✐♦♥✳

❙✐♥❝❡u✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥✱ ✇❡ ❦♥♦✇ t❤❛t Z T

S

Z

Ω

∂_t²u− △u

(m· ∇u)dx dt= 0

❆ ♥❡✇ ❢❛♠✐❧② ♦❢ s❡❝♦♥❞✲♦r❞❡r ❆❇❈s ❢♦r t❤❡ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ✲ P❛rt ■■✶✻

t❤❛t ✐s t♦ s❛② t❤❛t Z

Ω

∂tu(m· ∇u)dx T

S − Z T

S

Z

Ω

∂tu(m· ∇∂tu)dx dt+ Z T

S

Z

Ω∇u· ∇(m· ∇u)dx dt

− Z T

S

Z

∂Ω

∂nu(m· ∇u)dσ dt= 0

▼♦r❡♦✈❡r✱ ✇❡ ❝❛♥ ❝❤❡❝❦ t❤❛t ✐♥Ω

∂tu(m· ∇∂tu) = 1

2m· ∇|∂tu|² s♦ t❤❛t

Z

Ω

∂tu(m· ∇∂tu) = 1 2

Z

Ω

m· ∇|∂tu|²dx

=1 2

Z

∂Ω

m·n|∂tu|²dσ−1 2

Z

Ω

❞✐✈m|∂tu|²dx

❚❤❡r❡❢♦r❡✱ ✇❡ ❣❡t Z

Ω

∂tu(m· ∇u)dx T

S

−1 2

Z T S

Z

∂Ω

(m·n)|∂tu|²dσ dt+1 2

Z T S

Z

Ω

❞✐✈ m|∂tu|²dx dt+

Z T S

Z

Ω∇u· ∇(m· ∇u)dx dt− Z T

S

Z

∂Ω

∂nu(m· ∇u)dσ dt= 0,

✇❤✐❝❤ ❡♥❞s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧❡♠♠❛✳

✹✳✷ Pr♦♦❢ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❡♥❡r❣② ❞❡❝❛②

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ s❡t m(x) = x−x0 ✇❤❡r❡ x0 ∈ R^N ✭N = 2,3 ❞❡♥♦t❡s t❤❡

s♣❛❝❡ ❞✐♠❡♥s✐♦♥✮✳ ❲❡ s✉♣♣♦s❡ t❤❛t x0 ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t

Γ ={x∈∂Ω, m.n≤0} ✭✹✳✶✶✮

❛♥❞

Σ ={x∈∂Ω, m.n≥0}. ✭✹✳✶✷✮

❚❤✐s ❤②♣♦t❤❡s✐s ✐s s❛t✐s✜❡❞ ✐❢Ω✐s st❛r✲s❤❛♣❡❞ ✇✐t❤ r❡s♣❡❝t t♦x0✳

■♥ t❤❛t ❝❛s❡✱ ✇❡ ❦♥♦✇ t❤❛t ❞✐✈m=N✳ ❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ ✇❡ s✉♣♣♦s❡

t❤❛t N= 3 ❜✉t t❤❡r❡ ✐s ♥♦ ❞✐✣❝✉❧t② t♦ ♦❜t❛✐♥ t❤❡ s❛♠❡ r❡s✉❧t ❢♦rN= 2✳

❚♦ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥tC t❤❛t s❛t✐s✜❡s ✭✹✳✷✮✱ ✇❡ ♦♥❧②

❤❛✈❡ t♦ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢

1 2

Z T S

Z

Ω |∂tu|²+|∇u|² dx dt s✐♥❝❡ ✇❡ ❛❧r❡❛❞② ❦♥♦✇ ❢r♦♠ ▲❡♠♠❛ ✹✳✷ t❤❛t

Z T S

Z

Σ

κ

2|ψ|²dσ dt≤ 1 α_♠✐♥E(S).